# Questions tagged [legendre-polynomials]

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34
questions

**2**

votes

**2**answers

120 views

### A recurrence formula for the Legendre function $P_\mu^\nu(x)$

Im looking for a recurrence formula of type:
$$(\mu-\nu) x P_\mu^\nu(x) + P_{\mu-1}^\nu(x)=?, \quad \mu,\nu\in \mathbb R$$
where $P_\mu^\nu(x)$ is the Legendre function of the first kind (solution to ...

**5**

votes

**0**answers

77 views

### $L^p$-convergence of truncated Legendre series approximation to inverse error function $\text{erfinv}$

Bounding the $L^p$-error for an $n$-th order Legendre series approximation
I have function $f\,\colon (-1, 1) \to \mathbb{R}$ where $f \in L^q(-1, 1)$ for any $q \geq 1$. I approximate $f$ using a ...

**0**

votes

**0**answers

51 views

### Any good references on the decay rate of Legendre coefficient?

Let $P_n:[-1,1]\rightarrow\mathbb{R}$ be the $n$-th Legendre Polynomial. and, let
$$a_n:=\int_{-1}^1 f(t) P_n(t)\, dt$$
for some $f:[-1,1]\rightarrow\mathbb{R}$.
Are there any good references on the ...

**4**

votes

**1**answer

208 views

### Gegenbauer's addition theorem for Jacobi polynomials

I have the following identity,
$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z)
= 2 \, j_n(x) \, j_n(y) \;,$$
where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...

**1**

vote

**1**answer

144 views

### Generate a two-variable polynomial from its "roots [closed]

I know, from mathematics basis, that a polynomial with one variable can be factor in function of its roots, so I can generate any one-variable polynomial from its zeros.
But I want know if is ...

**5**

votes

**1**answer

337 views

### Convergence of the series of Legendre polynomials

Consider the generating function of Legendre polynomials:
$$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum\limits^{\infty}_{n=0} P_n(x)t^n$$
Is it true that for $0<x<1, t=1$ series of Legendre ...

**0**

votes

**0**answers

62 views

### Orthogonal functions and linear operators

Consider the following function $f: [-1,1] \rightarrow \mathbb{R}$, expanded in terms of Legendre functions,
$$
f(y;\boldsymbol{\beta}) = \sum_{i=0}^{\infty} \beta_i P_i(y)
$$
where $\boldsymbol{\beta}...

**4**

votes

**3**answers

276 views

### Legendre Polynomial Integral over half space

I need to compute the following integral
$$
I_{n,m} := \int_0^1 P_n(x) P_m(x) \; \mathrm{d}x
$$
where $P_n$ is the Legendre polynomial.
For an even sum $n+m=2l$ it is easy to show that
$$
I_{n,m} = \...

**4**

votes

**1**answer

154 views

### Integral with Legendre polynomial

Let $n$ be an integer $\geq 2$ and let $p_n(x)=\displaystyle \frac{1}{n!}(x^n(1-x)^n)^{(n)}$ be a Legendre polynomial.
Let $k $ be a positive integer.
I would like know if there is a closed formula (...

**3**

votes

**1**answer

145 views

### Looking for bound in integral involving Legendre polynomial

I'm looking for an upper bound to the following integral or equivalent when $n$ leads to $ +\infty $ to the following expression
$$I_n:=\left|\int_{0}^1 \int_{0}^1 \frac{p_n(x) p_n(y)}{(1-xy)} dx dy ...

**1**

vote

**2**answers

250 views

### Integral formula involving Legendre polynomial

I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...

**3**

votes

**2**answers

308 views

### Legendre equation: An interpretation [closed]

I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation:
\begin{eqnarray}
(1-x^2)y''-2xy'+l(l+1)y=0.
\end{eqnarray}
Doing some calculations, we ...

**2**

votes

**1**answer

173 views

### Integration of hypergeometric product for legendre polynomials

I'm looking for a general solution to the integral:
$\int_{0}^1 {_2}F_1(m-n,m+n+1,m+1;z){_2}F_1(m-k+1,m+k+2,m+2;z) dz$
where $m,n,k\in \mathbb{N}$ and $m\leqslant n$ and $m+1 \leqslant k$.
To give ...

**2**

votes

**0**answers

61 views

### Rate of convergence of generalized polynomial chaos

Let $\eta=g(\xi_1,\ldots,\xi_M)$ be a random variable expressed as a function of the random vector $\xi=(\xi_1,\ldots,\xi_M)$. Assume that $\xi_1,\ldots,\xi_M$ are absolutely continuous and ...

**3**

votes

**0**answers

108 views

### Identity for the product of two different associated Legendre polynomials

In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated:
$$
\small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}...

**2**

votes

**1**answer

418 views

### Clausen’s identity for associated Legendre polynomials

Clausen’s identity for Legendre polynomials has the form (see, for example,
A generating function of the squares of Legendre polynomials, by Wadim Zudilin: https://arxiv.org/abs/1210.2493)
$$P_n(\cos{\...

**3**

votes

**1**answer

400 views

### Bounds on Legendre polynomials on the complex plane

Let $P_n$ be the Legendre polynomial of degree $n$. Could you suggest me some references to bound the polynomials on the complex plane (near the real line in particular) ? More specifically, I need to ...

**5**

votes

**2**answers

1k views

### Relation between Legendre and Chebyshev polynomials

Where I could find relationships between Legendre and Chebyshev polynomials?
For example I found with maple
$$ P_n(\cos\theta)=\sum_{k=0}^n(-1)^{n+k}\frac{2-\delta_{k0}}{4^n}
\binom{n-k}{\frac{n-k}{2}...

**3**

votes

**0**answers

331 views

### Formula of Laplace for the asymptotic expansion of Legendre polynomials

According to the so-called formula of Laplace (see, e.g., [1, Theorem 8.21.2]), Legendre polynomials admit the following asymptotic expansion:
$$
P_n(\cos\theta) = \sqrt{2} (\pi n\sin\theta)^{-\frac12}...

**1**

vote

**0**answers

350 views

### Legendre functions as Hypergeometric functions

Is there some approximation or regularization that goes in tacitly in the following equality:
$Q_{\lambda }(z) = \frac{\sqrt{\pi } \Gamma (\lambda +1)}{2^{\lambda +1} \Gamma \left(\lambda +\frac{3}{2}...

**2**

votes

**0**answers

41 views

### Legendre expansion of $r(x) = f(x)/g(x)$ using a finite number of samples from $f(x)$ and $g(x)$

I have two finite sets of events $\{x_1, ..., x_N\}$ and $\{y_1, ..., y_N\}$ that are sampled from the PDFs $f(x)$ and $g(x)$, respectively, where $x \in [-1,+1]$. I want to estimate the Legendre ...

**1**

vote

**0**answers

152 views

### Expansion of prolate spheroidal harmonics

For two coordinate frames $O'$ and $O''$ both offset along the $z$-axis by $\pm R$ respectively, with corresponding offset spherical coordinates $r'$, $\theta'$, $r''$ and $\theta''$, and with prolate ...

**5**

votes

**2**answers

534 views

### Recurrence of Legendre polynomial roots/ quadrature points

Consider Legendre polynomials $p_n (x)$ on $[-1,1]$. For each $n \in \mathbb{N}$ we denote the zeros of $p_n (x)$ by $\left( x_j ^{(n)} \right) _{j=1} ^n$.
We know that these roots are distinct, and ...

**0**

votes

**1**answer

822 views

### Integrals involving associated legendre polynomials

Do the following integrals have a closed-form solution for any integer value of $m,l,k$ and $n$?
$\int^{\pi}_{0} P^{m}_{l}\left(\cos\theta\right)P^{n}_{k}\left(\cos\theta\right)\cot\theta d\theta$
$\...

**7**

votes

**2**answers

3k views

### Legendre Polynomial Integral

How can I evaluate
$$ \int_{-1}^1 P_n(x)P_l(x)x^k dx $$
when $k$ is even?
Or what might be a source where I could find integrals like this?

**6**

votes

**2**answers

400 views

### Symmetric matrix formula for Gauss-Legendre quadrature

While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...

**4**

votes

**1**answer

538 views

### Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate.
Longer: If $p_n$ is the $n$-th Legendre polynomial, ...

**1**

vote

**0**answers

323 views

### Legendre equation with homogeneous boundary condition

I'm looking for solutions to the Legendre differential equation
$$
\frac{d}{dx}\left((1-x^2)\frac{d}{dx}f(x)\right)+(\alpha-1)\alpha f(x)=0
$$
with boundary conditions $f'(0)=0$ and $f(1)=0$, but ...

**15**

votes

**4**answers

788 views

### Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant
$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i)
$$
over all $a_0,\dots,a_{n-1}$ such ...

**14**

votes

**2**answers

832 views

### Computing Gauss Legendre quadrature for large $N$

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...

**0**

votes

**1**answer

125 views

### Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]

By playing around with assoc. Legendre polynomials, I arrived at
$$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$
Now, I want to show that we don't have equality ...

**0**

votes

**1**answer

356 views

### Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared:
$$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$
where $\rho,s>1$, $Q^{\...

**6**

votes

**0**answers

264 views

### Legendre polynomials and formal groups

Let $P_n(x)$ be Legendre polynomials:
$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$
Usual arguments from the theory of formal groups allow to
prove that for any $n$
$$P_n(x)=Q_n(...

**1**

vote

**0**answers

119 views

### Fourier-Legendre coefficients and Sobolev regularity

Let $f$ be an $L^1$-function supported in $[-b, b]$, where $0 < b \le b_0 < 1$. Let $Q_n$ be the normalised Legendre polynomials and let
$a_n = \int f(x) Q_n(x) dx$ be the Fourier-Legendre ...